$(i)$ For the wave $y(x, t) = 0.06 \sin \left(\frac{2 \pi}{3} x\right) \cos (120 \pi t)$,where $x$ and $y$ are in $m$ and $t$ is in $s$. The length of the string is $1.5 \, m$ and its mass is $3.0 \times 10^{-2} \, kg$. Do all the points on the string oscillate with the same $(a)$ frequency,$(b)$ phase,$(c)$ amplitude? Explain your answers.
$(ii)$ What is the amplitude of a point $0.375 \, m$ away from one end?

(N/A) $(i)$ The given equation represents a standing wave.
$(a)$ Yes,all points on the string oscillate with the same frequency of $f = \frac{\omega}{2\pi} = \frac{120\pi}{2\pi} = 60 \, Hz$,except at the nodes where the amplitude is zero.
$(b)$ No,all points do not oscillate with the same phase. Points within the same loop oscillate in phase,but points in adjacent loops oscillate with a phase difference of $\pi$ radians.
$(c)$ No,all points do not oscillate with the same amplitude. The amplitude of a point at position $x$ is given by $A(x) = 0.06 \sin \left(\frac{2\pi}{3} x\right)$,which depends on the position $x$.
$(ii)$ The amplitude of a point at $x = 0.375 \, m$ is:
$A = 0.06 \sin \left(\frac{2\pi}{3} \times 0.375\right)$
$A = 0.06 \sin \left(\frac{2\pi}{3} \times \frac{3}{8}\right)$
$A = 0.06 \sin \left(\frac{\pi}{4}\right)$
$A = 0.06 \times \frac{1}{\sqrt{2}} \approx 0.0424 \, m$.